Start small. Given $M = m_1 m_2$, can you think of another message $M'$ such that $S(M) = S(M')$? (Note that a weakness could be more complicated than this. For example, it could be impossible to find two messages with the same signature, but possible to derive the signature of $M'$ from the signature of $M$. However, in this case, it is possible to forge another message with the same signature.)

Above I used the fact that $\oplus$ is commutative. What other algebraic properties does $\oplus$ have?

The normal way to sign a message $m_1 \ldots m_n$ is of course $\textsf{RSASSA}(h(m_1 \ldots m_n))$. If you also want to be able to verify the signature of one part independently, one solution is to build a hash tree and sign the root hash. A two-level hash tree would be $h(h(m_1) \ldots h(m_n))$. If you transmit the signature $\textsf{RSASSA}(h(h(m_1) \ldots h(m_n)))$ as well as the list of individual hashes $(h(m_1), \ldots, h(m_n))$ then it's possible to verify the signature of any of the individual $m_i$. Signatures are larger than hashes and more expensive to compute, so this can save resources compared so signing each $m_i$ independently.